In this study, we will focus on qubit-based QEC with trapped ions. To date, trapped-ion experiments have shown high fidelity gates and long coherence times [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, making them one of the most promising candidates for the future realisation of large-scale quantum computers. There are currently two approaches for the scalability of trapped-ion systems for QEC. The first approach is the single-string ion trap, where all the ions are placed in modules consisting of a single, linear radio-frequency trap, and coupled by photonic interconnects [8]}, [9]}. On the other hand, there is a so-called shuttling-based approach to scalability [10]}, which utilises microfabricated traps. In this approach, the ions are moved between storage and manipulation zones, where they are subjected to single-qubit gates and fluorescence measurements as needed or otherwise merged with other ions to implement entangling gates or sympathetic laser cooling. After applying the desired sequence of operations, the ions can be split and shuttled back to the storage zones. There they remain as spectators of the operations on other active ions in the manipulation zones [11]}, [12]}, [13]}, [10]}, [15]}, [16]}, [17]}. Our study is motivated by new experimental capabilities that have recently emerged in the single-string trapped-ion modules. These new experimental capabilities include the possibility to perform high-fidelity entangling gates addressed on specific subsets of ions [18]}, [19]}, [20]} , leading to effective all-to-all connectivity for two-qubit entangling gates. These capabilities have motivated the study of the single-string ion trap scenario as a viable approach to implementing an operational logical qubit [21]}, [11]}. Informed by these results, we focus our study on the single string ion traps, which can eventually be combined with the shuttling-based or photonic-interconnect techniques for further scalability.
